Historically bust is derived from a dialect pronunciation of burst and is related to it much as cuss is related to curse.Bust is both a noun and a verb and has a wide range of meanings for both uses. Many are slang or informal. A few, as “a decline in economic conditions, depression,” are standard.

bust

n.

1690s, "sculpture of upper torso and head," from French buste (16c.), from Italian busto "upper body," from Latin bustum "funeral monument, tomb," originally "funeral pyre, place where corpses are burned," perhaps shortened from ambustum, neuter of ambustus "burned around," past participle of amburere "burn around, scorch," from ambi- "around" + urere "to burn." Or perhaps from Old Latin boro, the early form of classical Latin uro "to burn." Sense development in Italian is probably from Etruscan custom of keeping dead person's ashes in an urn shaped like the person when alive. Meaning "bosom" is by 1884.

variant of burst (n.), 1764, American English. For loss of -r-, cf. ass (n.2). Originally "frolic, spree;" sense of "sudden failure" is from 1842. Meaning "police raid or arrest" is from 1938. Phrase ______ or bust as an emphatic expression attested by 1851 in British depictions of Western U.S. dialect. Probably from earlier expression bust (one's) boiler, by late 1840s, a reference to steamboat boilers exploding when driven too hard.

v.

"to burst," 1806, variant of burst (v.); for loss of -r-, cf. ass (n.2). Meaning "go bankrupt" is from 1834. Meaning "break into" is from 1859. The slang meaning "demote" (especially in a military sense) is from 1918; that of "place under arrest" is from 1953 (earlier "to raid" from Prohibition). In card games, "to go over a score of 21," from 1939. Related: Busted; busting.